Quasilinearization for resonant boundary-value problems with mixed boundary conditions
We consider resonant problems of the type (i) x′′ + p(t)x′ + q(t)x = f (t, x, x′), (ii) x′(0) = 0, x(T) = 0, where p, q, f are continuous functions and the homogeneous problem (iii) x′′ + p(t)x′ + q(t)x = 0 with boundary-value conditions (ii) has a nontrivial solution. We study this problem by modif...
Gespeichert in:
| Datum: | 2014 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2014
|
| Schriftenreihe: | Нелінійні коливання |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/175859 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Quasilinearization for resonant boundary-value problems with mixed boundary conditions / N. Sveikate, F. Sadyrbaev // Нелінійні коливання. — 2014. — Т. 17, № 1. — С. 112-126. — Бібліогр.: 7 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-175859 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1758592021-02-03T01:30:17Z Quasilinearization for resonant boundary-value problems with mixed boundary conditions Sveikate, N. Sadyrbaev, F. We consider resonant problems of the type (i) x′′ + p(t)x′ + q(t)x = f (t, x, x′), (ii) x′(0) = 0, x(T) = 0, where p, q, f are continuous functions and the homogeneous problem (iii) x′′ + p(t)x′ + q(t)x = 0 with boundary-value conditions (ii) has a nontrivial solution. We study this problem by modifying the linear part and applying the quasilinearization technique to the modified problem. Розглянуто задачi типу (i) x′′ + p(t)x′ + q(t)x = f (t, x, x′), (ii) x′(0) = 0, x(T) = 0, з резонансом, де p, q, f — неперервнi функцiї та однорiдна задача (iii) x′′ + p(t)x′ + q(t)x = 0 разом з граничними умовами (ii) має нетривiальний розв’язок. Задача вивчається за допомогою змiни лiнiйної частини та застосування технiки квазiлiнеаризацiї до модифiкованої задачi. 2014 Article Quasilinearization for resonant boundary-value problems with mixed boundary conditions / N. Sveikate, F. Sadyrbaev // Нелінійні коливання. — 2014. — Т. 17, № 1. — С. 112-126. — Бібліогр.: 7 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/175859 517.9 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We consider resonant problems of the type (i) x′′ + p(t)x′ + q(t)x = f (t, x, x′), (ii) x′(0) = 0, x(T) = 0, where p, q, f are continuous functions and the homogeneous problem (iii) x′′ + p(t)x′ + q(t)x = 0 with boundary-value conditions (ii) has a nontrivial solution. We study this problem by modifying the linear part and applying the quasilinearization technique to the modified problem. |
| format |
Article |
| author |
Sveikate, N. Sadyrbaev, F. |
| spellingShingle |
Sveikate, N. Sadyrbaev, F. Quasilinearization for resonant boundary-value problems with mixed boundary conditions Нелінійні коливання |
| author_facet |
Sveikate, N. Sadyrbaev, F. |
| author_sort |
Sveikate, N. |
| title |
Quasilinearization for resonant boundary-value problems with mixed boundary conditions |
| title_short |
Quasilinearization for resonant boundary-value problems with mixed boundary conditions |
| title_full |
Quasilinearization for resonant boundary-value problems with mixed boundary conditions |
| title_fullStr |
Quasilinearization for resonant boundary-value problems with mixed boundary conditions |
| title_full_unstemmed |
Quasilinearization for resonant boundary-value problems with mixed boundary conditions |
| title_sort |
quasilinearization for resonant boundary-value problems with mixed boundary conditions |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/175859 |
| citation_txt |
Quasilinearization for resonant boundary-value problems with mixed boundary conditions / N. Sveikate, F. Sadyrbaev // Нелінійні коливання. — 2014. — Т. 17, № 1. — С. 112-126. — Бібліогр.: 7 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT sveikaten quasilinearizationforresonantboundaryvalueproblemswithmixedboundaryconditions AT sadyrbaevf quasilinearizationforresonantboundaryvalueproblemswithmixedboundaryconditions |
| first_indexed |
2025-07-15T13:17:51Z |
| last_indexed |
2025-07-15T13:17:51Z |
| _version_ |
1837719063793500160 |
| fulltext |
UDC 517.9
QUASILINEARIZATION FOR RESONANT BOUNDARY-VALUE
PROBLEMS WITH MIXED BOUNDARY CONDITIONS*
КВАЗIЛIНЕАРИЗАЦIЯ ДЛЯ ГРАНИЧНИХ ЗАДАЧ З РЕЗОНАНСОМ
ТА МIШАНИМИ ГРАНИЧНИМИ УМОВАМИ
N. Sveikate
Daugavpils Univ.
Parades str., 1, Daugavpils, LV 5401, Latvia
e-mail: nsveikate@inbox.lv
F. Sadyrbaev
Univ. Latvia
Raina blvd., 19, Riga, LV 1586, Latvia
e-mail: felix@latnet.lv
We consider resonant problems of the type (i) x′′ + p(t)x′ + q(t)x = f(t, x, x′), (ii) x′(0) = 0, x(T ) = 0,
where p, q, f are continuous functions and the homogeneous problem (iii) x′′ + p(t)x′ + q(t)x = 0
with boundary-value conditions (ii) has a nontrivial solution. We study this problem by modifying the li-
near part and applying the quasilinearization technique to the modified problem.
Розглянуто задачi типу (i) x′′ + p(t)x′ + q(t)x = f(t, x, x′), (ii) x′(0) = 0, x(T ) = 0, з резонансом,
де p, q, f — неперервнi функцiї та однорiдна задача (iii) x′′ + p(t)x′ + q(t)x = 0 разом з гранични-
ми умовами (ii) має нетривiальний розв’язок. Задача вивчається за допомогою змiни лiнiйної
частини та застосування технiки квазiлiнеаризацiї до модифiкованої задачi.
1. Introduction. It is well known that the two-point boundary-value problem (BVP)
x′′ + px′ + qx = F (t, x, x′), x′(0) = 0, x(T ) = 0, (1.1)
where p, q ∈ C(R) and F ∈ C([0, T ] × R × R,R) is a bounded function, has a C2-solution if
the linear part (l2x)(t) := x′′ + px′ + qx is nonresonant with respect to the above boundary
conditions. In other words, if the linear homogeneous problem
x′′ + px′ + qx = 0, x′(0) = 0, x(T ) = 0,
has only the trivial solution, then the problem (1.1) is solvable.
We consider the problem
x′′ + px′ + qx = f(t, x, x′), x′(0) = 0, x(T ) = 0 (1.2)
∗ This work has been supported by the European Social Fund within the Project “Support for the implementati-
on of doctoral studies at Daugavpils University” Agreement Nr. 2009/0140/1DP/1.1.2.1.2/09/IPIA/VIAA/015.
c© N. Sveikate, F. Sadyrbaev, 2014
112 ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
QUASILINEARIZATION FOR RESONANT BOUNDARY-VALUE PROBLEMS . . . 113
with a resonant linear part, where the right-hand side f may be unbounded.
Problems of the type (1.2) arise when considering centrally symmetric solutions of elliptic
partial differential equations. The center of symmetry (the origin) corresponds to the first
boundary condition in (1.2).
The conditions for existence of a solution to the problem (1.2) are provided in the work [3].
To get existence and multiplicity results, we use the quasilinearization approach elaborated
in the works [6, 7].
This method is based on the idea of replacing a given resonant problem with a nonresonant
one and proving some estimates that ensure that there exists a bounded domain D in the space
of variables (t, x, x′) such that both the resonant and the nonresonant problems are equivalent
in D.
The paper consists of the introduction, the definitions section, the section with Main Theo-
rem and two sections where respectively nonresonant and resonant boundary-value problems
are considered for a Emden – Fowler type equation.
2. Definitions.
Definition 2.1. We say that the linear part (l2x)(t) is i-nonresonant with respect to the mixed
boundary conditions x′(0) = 0, x(T ) = 0 if a solution x(t) of the Cauchy problem
(l2x)(t) = 0, x′(0) = 0, x(0) = 1
has exactly i zeros in the interval (0, T ) and x(T ) 6= 0.
For instance, the linear expression x′′ + 10x is 1-nonresonant in the interval (0, 1), since a
solution of the respective Cauchy problem has exactly one zero in the interval.
Definition 2.2. We say that a solution ξ(t) of the BVP (1.2) is of type i, i = 0, 1, . . . , if
the difference x(t;α) − ξ(t) for sufficiently small α has exactly i zeros in the interval (0, T ) and
x(T ;α)− ξ(T ) 6= 0, where x(t;α) is a solution of
x′′ + px′ + qx = f(t, x, x′), x′(0) = ξ′(0) = 0, x(0) = ξ(0) + α. (2.1)
Remark 2.1. Definition 2.2 fits for the case of f(t, x, x′) in (2.1) being merely continuous. If
f has continuous partial derivatives fx and fx′ , the equation of variations
y′′ + p(t)y′ + q(t)y = fx
(
t, ξ(t), ξ′(t)
)
y + fx′
(
t, ξ(t), ξ′(t)
)
y′, y′(0) = 0, y(0) = 1, (2.2)
can be considered and the definition then becomes as follows: a solution ξ(t) is of type i if y(t)
in (2.2) has exactly i zeros in (0, T ) and y(T ) 6= 0.
3. Main theorem. Consider the quasilinear problem (1.1), where the right-hand side F is
bounded.
To the end of this section we assume that the linear part in (1.1) is nonresonant.
Lemma 3.1. If the continuous function F in (1.1) is bounded, then problem (1.1) has a solu-
tion.
This is a well-known result that follows from Conti’s theorem [2].
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
114 N. SVEIKATE, F. SADYRBAEV
Lemma 3.2. A set S of all solutions of problem (1.1) is compact in C1([0, T ]).
This can be proved by a routine application of the Arzela – Ascoli criterion using the Green’s
function representation of a solution to the BVP.
Corollary 3.1. A set of initial data {x(0)} of all solutions to the BVP (1.1) is compact in R.
Proof. Follows from compactness of the set S of all solutions of problem (1.1).
Consider solutions x(t; γ) of the Cauchy problem
x′′ + px′ + qx = F (t, x, x′), x′(0) = 0, x(0) = γ. (3.1)
Lemma 3.3. Let ξ(t) be any solution of problem (1.1), where the linear part (l2x)(t) is i-
nonresonant.
Then for α large enough the difference u(t, α) = x(t;α) − ξ(t) has exactly i zeros in (0, T )
and u(T ;α) 6= 0.
Essentially Lemma 2.3 in [6].
The next result on separation of zeros is of Valle Poussin type [5] (Chapter 3, Section 17).
Lemma 3.4. Let ξ(t) be any solution of problem (1.1), where the linear part (l2x)(t) is nonre-
sonant, and u(t;α) is as in Lemma 3.3.
Zeros ti(α) (if any) of the function u(t;α) continuously depend on α. If |α| < B < +∞, then
there exists δ(B) > 0 such that the distance between two consecutive zeros of u(t;α) cannot be
less that δ. If u(T ;α0) = 0 for some α0 6= ξ(0) then the respective x(t;α0) solves problem (1.1).
Essentially Lemma 2.4 in [6].
Theorem 3.1. A quasilinear problem (1.1) with i-nonresonant linear part has a solution x(t)
of type i.
Proof. Consider a set S of all solutions of problem (1.1). This set is compact in C1([0, T ])
and a set of the initial conditions x(0) of elements of S is compact in R. Therefore there exist
solutions xmax(t) and xmin(t) with the maximal and minimal values of x(0) respectively. In case
of uniqueness of a solution these solutions coincide.
Consider a solution xmax(t) with the property that xmax(0) = maxS{x(0)}. We claim that
this solution possesses the property of being a solution of type i. Suppose this is not the case.
Consider solutions x(t; γ) of the Cauchy problem for γ > xmax(0). For γ tending to +∞ the
functions u(t; γ) = x(t; γ)− xmax(t) have exactly i zeros and u(T ; γ) 6= 0, by Lemma 3.3.
Consider the following cases.
Case 1. There exists a sequence {γn}, γn > xmax(0), such that the differences x(t; γn) −
−xmax(t) have more than i zeros in (0, T ]. If x(T ; γn)− xmax(T ) = 0 for some n, then a soluti-
on x(t; γn) solves problem (1.1). This is in contradiction with the assumption that xmax(t) is a
solution of problem (1.1) with the maximal initial value x(0).Therefore we may suppose that the
differences x(t; γn)−xmax(t) have more than i zeros in the open interval (0, T ).Denote ti+1(γn)
the (i + 1)-st zero of the differences x(t; γn) − xmax(t). By Lemma 3.4, it is separated from
neighboring zeros ti(γn) and, if any, from ti+2(γn). Moreover, ti+1(γ) changes continuously
together with γ = x(0). Thus there exists γ∗ > xmax(0) such that the respective solution
x(t; γ∗) has its (i + 1)-st zero at t = T. Therefore this solution of the equation solves also the
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
QUASILINEARIZATION FOR RESONANT BOUNDARY-VALUE PROBLEMS . . . 115
BVP (1.1). The contradiction with the maximality property of xmax(t) is obtained. Therefore
the considered case is not possible.
Case 2. There exists a sequence {γn}, γn > xmax(0), such that the differences x(t; γn) −
−xmax(t) have less than i zeros in (0, T ]. If x(T ; γn)− xmax(T ) = 0 for some n, then a solution
x(t; γn) solves the problem (1.1). The contradiction follows as in the above case. Suppose that
the differences x(t; γn) − xmax(t) have less than i zeros in the open interval (0, T ). Denote
by ti−1(γn) the (i − 1)-th zero (if i ≥ 1) of the differences x(t; γn) − xmax(t). It is separated
from neighboring zeros if γ ∈ (xmax(0),Γ), where Γ is a large positive number such that a
solution x(t; Γ) of the Cauchy problem (3.1) (where γ = Γ) has exactly i zeros in (0, T ) and
does not vanish at t = T. An extra zero ti(γ) has to appear in the interval (0, T ] then. Suppose
a solution x(t; γ∗) is such a solution. Then it solves problem (1.1) and, therefore, a contradiction
is obtained with the maximality property of xmax(t).
If i = 0, the consideration may be conducted similarly.
Theorem 3.1 is proved.
4. Multiple solutions. The results of the previous section allow for treating multiple solutions
of the BVP (1.2). Indeed, suppose that the problem can be reformulated so that the right-hand
side is bounded and the modified problem hence has a solution. If the graph (t, x(t), x′(t)) of
this solution lies in the domainDi where the original and the modified equations are equivalent,
then x(t) solves also the original problem. Hence the existence. Moreover, if the linear part of
the modified problem is i-nonresonant then there exists a solution x(t) of the same type i.
Suppose that this process can be repeated with some other linear part of type j 6= i. Then
it can be concluded that the original problem also has a solution of type j. Hence the second
conditions.
In the sequel we will show that this approach is possible for both the nonresonant and even
the resonant the problems.
To be specific, we choose to treat the Emden – Fowler type equations.
5. Mixed problem. Nonresonant case. Consider the problem
x′′ + r2x = −Q(t)|x|psignx, (5.1)
x′(0) = 0, x(1) = 0, (5.2)
where the linear part x′′ + r2x is nonresonant with respect to the boundary conditions, that is
r 6= π
2
+ πn, n = 0, 1, 2, . . . .
We show that it is possible to quasilinearize equation in (5.1) so that multiple solutions are
revealed in the spirit of Main Theorem 3.1.
Theorem 5.1. Suppose that
0 < q1 ≤ Q(t) ≤ q2 (5.3)
and the inequality
k2√
k2 + r2 cos
√
k2 + r2
< β
p
p
p−1
|p− 1|
(
q1
q2
) 1
|p−1|
(5.4)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
116 N. SVEIKATE, F. SADYRBAEV
Fig 5.1. Existence of a number Nk.
holds for some k2 ∈
(
π2(2i− 1)2
4
− r2, π
2(2i+ 1)2
4
− r2
)
, i = 1, 2, . . . , where β > 1 is a root
of the equation
βp = β + (p− 1)p
p
1−p . (5.5)
Then there exists an i-type solution of problem (5.1), (5.2).
Proof. We use the quasilinearization approach.
First we modify equation (5.1) adding a linear part
x′′ + r2x+ k2x = k2x−Q(t)|x|psignx.
The linear part (L2x)(t) := x′′+ (r2 + k2)x is nonresonant with respect to the boundary condi-
tions (5.2).
We define
fk(t, x) := k2x−Q(t)|x|psignx. (5.6)
We wish to bound the right-hand side in (5.6). Since it is odd in x for fixed t, let us consider
it for nonnegative values of x. An extremum of fk(t, x) is attained (maximum for p > 1 and
minimum for 0 < p < 1) at the point
x0 =
(
k2
pQ(t)
) 1
p−1
.
We can calculate the value of the function at the point of maximum x0, see Fig. 5.1. Set
Mk(t) = |fk(t, x0)| =
(
k2
p
) p
p−1
|p− 1|Q
1
1−p .
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
QUASILINEARIZATION FOR RESONANT BOUNDARY-VALUE PROBLEMS . . . 117
We choose constants Nk so that
∀(t, x) : t ∈ [0, 1], |x| ≤ Nk ⇒ |fk(t, x)| ≤ Mk.
The value of Nk(t) can be computed by solving the equation
fk(t, x) = −fk(t, x0),
or, equivalently, that of
k2x−Q(t)xp =
(
k2
p
) p
p−1
|p− 1|Q
1
1−p
with respect to x for any fixed t. Computation gives that
Nk(t) =
(
k2
Q(t)
) 1
p−1
β,
where the constant β is a positive root of the equation (5.5).
Then let us consider the quasilinear equation
x′′ + (r2 + k2)x = Fk(t, x), (5.7)
where Fk(t, x) = fk(δ(−Nk, x,Nk)) and δ is a “cut-of” function, that is,
δ(−Nk, x,Nk) =
Nk, x > Nk,
x, −Nk ≤ x ≤ Nk,
−Nk, x < Nk,
and max
{
|Fk| : x ∈ R
}
= Mk.
The original problem (5.1), (5.2) and the quasilinear one (5.7), (5.2) are equivalent in the
domain Ωk = {(t, x) : 0 ≤ t ≤ 1, |x(t)| ≤ Nk} . Problem (5.7), (5.2) can be written in the
integral form,
xk(t) =
1∫
0
Gk(t, s)Fk(s, x(s))ds,
where Gk(t, s) is Green’s function [1, 4] to the respective homogeneous problem
x′′ + (r2 + k2)x = 0, x′(0) = x(1) = 0.
It is given by
Gk(t, s) =
sin
√
r2 + k2(s− 1) cos
√
r2 + k2t√
r2 + k2 cos
√
r2 + k2
, 0 ≤ t ≤ s ≤ 1,
sin
√
r2 + k2(t− 1) cos
√
r2 + k2s√
r2 + k2 cos
√
r2 + k2
, 0 ≤ s < t ≤ 1,
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
118 N. SVEIKATE, F. SADYRBAEV
and satisfies the estimate
|Gk(t, s)| ≤ Γk =
1
√
r2 + k2
∣∣∣cos
√
r2 + k2
∣∣∣ . (5.8)
It follows from (5.8) that
|x(t)| ≤ ΓkMk.
If the inequality
ΓkMk < Nk (5.9)
holds, then a solution x(t) of the quasilinear problem (5.7), (5.2) satisfies the estimate
|x(t)| < Nk ∀t ∈ [0, 1],
and the original problem (5.1), (5.2) allows for quasilinearization with respect to the domain
Ωk and the linear part (L2x)(t) := x′′ + (r2 + k2)x. It follows from Main Theorem 3.1 that if
the linear part (L2x)(t) is i-nonresonant, then problem (5.1), (5.2) has an i-type solution.
Consider inequality (5.9) and assume that Q(t) satisfies estimates (5.3). If p > 1, then
max
t∈[0,1]
Mk(t) =
(
k2
p
) p
p−1
|p− 1|q
1
1−p
1 ,
min
t∈[0,1]
Nk(t) =
(
k2
q2
) 1
p−1
β;
but in the case of 0 < p < 1 we have
max
t∈[0,1]
Mk(t) =
(
k2
p
) p
p−1
|p− 1|q
1
1−p
2 ,
min
t∈[0,1]
Nk(t) =
(
k2
q1
) 1
p−1
β.
Hence inequality (5.9) reduces to (5.4).
Theorem 5.1 proved.
Corollary 5.1. If there exist numbers k2 ∈
(
π2(2i− 1)2
4
− r2, π
2(2i+ 1)2
4
− r2
)
, i = 1, 2, . . .
. . . , n, which satisfy inequality (5.4), then there exist at least n solutions of different types to
problem (5.1), (5.2).
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
QUASILINEARIZATION FOR RESONANT BOUNDARY-VALUE PROBLEMS . . . 119
Table 5.1. Results of calculations for the nonresonant BVP x′′+π2x = −|x|psignx, x′(0) = 0 = x(1),
p > 1
p i Nonresonant intervals k ΓkMk Nk
3 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 0.456 1.814
2 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 0.465 2.978
2
(
π
√
5
2 ; π
√
21
2
)
3π
2 26.738 26.806
5
4 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 2.289 47.492
2
(
π
√
5
2 ; π
√
21
2
)
3π
2 95945.8 311593.0
3
(
π
√
21
2 ; π
√
45
2
)
5π
2 1.521 · 107 1.855 · 107
6
5 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 4.617 117.828
2
(
π
√
5
2 ; π
√
21
2
)
3π
2 1.742 · 106 6.958 · 106
3
(
π
√
21
2 ; π
√
45
2
)
5π
2 7.669 · 108 11.507 · 108
8
7 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 20.599 722.124
2
(
π
√
5
2 ; π
√
21
2
)
3π
2 6.296 · 108 34.539 · 108
3
(
π
√
21
2 ; π
√
45
2
)
5π
2 2.139 · 1012 14.408 · 1012
4
(
π
√
45
2 ; π
√
77
2
)
7π
2 4.601 · 1014 4.898 · 1014
21
20 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 9.462 · 105 917.88 · 105
2
(
π
√
5
2 ; π
√
21
2
)
3π
2 7.351 · 1025 111.594 · 1025
3
(
π
√
21
2 ; π
√
45
2
)
5π
2 1.464 · 1035 8.348 · 1035
4
(
π
√
45
2 ; π
√
77
2
)
7π
2 1.984 · 1041 5.844 · 1041
5
(
π
√
77
2 ; π
√
117
2
)
9π
2 7.575 · 1045 13.567 · 1045
6
(
π
√
117
2 ; π
√
165
2
)
11π
2 3.456 · 1049 4.154 · 1049
For instance, we consider the BVP
x′′ + π2x = −|x|psignx, x′(0) = 0, x(1) = 0. (5.10)
The linear part (L2x)(t) := x′′ + π2x is nonresonant with respect to the given boundary condi-
tions. To modify the differential equation we use values of k in the form
π
2
+πn, n = 0, 1, 2, . . . .
We obtained that for some values of p inequality (5.4) holds. The obtained results are given in
Tables 5.1 and 5.2.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
120 N. SVEIKATE, F. SADYRBAEV
Table 5.2. Results of calculations for the nonresonant BVP x′′ + π2x = −|x|psign x, x′(0) = 0 =
= x(1), 0 < p < 1
p i Nonresonant intervals k ΓkMk Nk
20
21 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 7.837 · 10−4 770.654 · 10−4
2
(
π
√
5
2 ; π
√
21
2
)
3π
2 7.577 · 10−30 70.431 · 10−30
3
(
π
√
21
2 ; π
√
45
2
)
5π
2 5.863 · 10−39 33.894 · 10−39
4
(
π
√
45
2 ; π
√
77
2
)
7π
2 8.273 · 10−45 24.703 · 10−45
5
(
π
√
77
2 ; π
√
117
2
)
9π
2 3.545 · 10−49 6.437 · 10−49
6
(
π
√
117
2 ; π
√
165
2
)
11π
2 1.155 · 10−52 1.407 · 10−52
7
8 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 2.693 · 10−5 97.983 · 10−5
2
(
π
√
5
2 ; π
√
21
2
)
3π
2 3.998 · 10−12 22.762 · 10−12
3
(
π
√
21
2 ; π
√
45
2
)
5π
2 3.002 · 10−15 6.421 · 10−15
4
(
π
√
45
2 ; π
√
77
2
)
7π
2 2.669 · 10−17 2.948 · 10−17
5
6 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 2.237 · 10−4 149.065 · 10−4
2
(
π
√
5
2 ; π
√
21
2
)
3π
2 2.690 · 10−9 11.302 · 10−9
3
(
π
√
21
2 ; π
√
45
2
)
5π
2 1.559 · 10−11 2.460 · 10−11
4
5 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 6.751 · 10−4 149.065 · 10−4
2
(
π
√
5
2 ; π
√
21
2
)
3π
2 7.306 · 10−8 25.244 · 10−8
3
(
π
√
21
2 ; π
√
45
2
)
5π
2 1.176 · 10−9 11.526 · 10−9
1
2 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 0.031 0.239
2
(
π
√
5
2 ; π
√
21
2
)
3π
2 0.002 0.003
1
3 1
(
−π
√
3
2 ; π
√
5
2
)
π
2 0.075 0.397
For instance, if p =
5
4
, there exist at least three values k0 =
π
2
, k1 =
3π
2
and k2 =
5π
2
,
which satisfy inequality (5.4). For example,
x′′ + π2x = −|x|
5
4 signx, x′(0) = 0, x(1) = 0,
has solutions of different types (see Fig. 5.2 – 5.4).
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
QUASILINEARIZATION FOR RESONANT BOUNDARY-VALUE PROBLEMS . . . 121
(a) (b)
Fig. 5.2. 1-Type solution of the problem (5.10): (a) the trivial solution of problem (5.10), (b) the di-
fference between the neighboring solution and the trivial solution of problem (5.10).
(a) (b)
Fig. 5.3. 2-Type solution of problem (5.10): (a) a solution of problem (5.10) given for the initial data
ε′2(0) = 0, ε2(0) = 30000, (b) the difference between the neighboring solution and the soluti-
on ε2(t) of problem (5.10).
(a) (b)
Fig. 5.4. 3-Type solution of the problem (5.10): (a) a solution of problem (5.10) given for the initial data
ε′3(0) = 0, ε3(0) = 9000000, (b) the difference between the neighboring solution and the solu-
tion ε3(t) of problem (5.10).
6. Mixed problem. Resonant case. Consider the problem
x′′ +
(π
2
)2
x = −Q(t)|x|psignx, (6.1)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
122 N. SVEIKATE, F. SADYRBAEV
x′(0) = 0, x(1) = 0, (6.2)
where the linear part x′′ + k2x is resonant with respect to the boundary conditions, since the
linear Cauchy problem
x′′ +
(π
2
)2
x = 0, x′(0) = 0, x(0) = 1,
has the nontrivial solution x(t) = cos
π
2
t.
We show that it is possible to quasilinearize equation in (6.1) so that multiple solutions are
revealed in the spirit of Main Theorem 3.1.
Theorem 6.1. Suppose that
0 < q1 ≤ Q(t) ≤ q2
and the inequality
2k2
√
π2 + 4k2 cos
√
π2+4k2
2
< β
p
p
p−1
|p− 1|
(
q1
q2
) 1
|p−1|
(6.3)
holds for some k ∈
(
π
√
i(i− 1), π
√
i(i+ 1)
)
, i = 1, 2, . . . , where β > 1 is a root of the
equation
βp = β + (p− 1)p
p
1−p .
Then there exists an i-type solution of problem (6.1), (6.2).
Proof. Now we modify equation (6.1) adding a linear part so that the resulting linear part
becomes nonresonant.
The proof is conducted as that of Theorem 5.1, replacing r2 with
(π
2
)2
in all formulas.
Corollary 6.1. If there exist numbers k ∈
(
π
√
i(i− 1), π
√
i(i+ 1)
)
, i = 1, 2, . . . , n, which
satisfy inequality (6.3), then there exist at least n solutions of different types to problem (6.1), (6.2).
We obtained the results (see Table 6.1) for certain values of p,which show that some numbers
k in the form
π
2
+πn, n = 0, 1, 2, . . . , for any respective nonresonance interval satisfy inequali-
ty (6.1).
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
QUASILINEARIZATION FOR RESONANT BOUNDARY-VALUE PROBLEMS . . . 123
Table 6.1. Results of calculations for the resonant BVP x′′+
π2
4
x = −|x|psign x, x′(0) = 0 = x(1)
p i Nonresonant intervals k ΓkMk Nk
3 1 (0;π
√
2) π
2 1.348 1.814
2 1 (0;π
√
2) π
2 1.131 2.978
5
4 1 (0;π
√
2) π
2 5.568 47.492
6
5 1 (0;π
√
2) π
2 11.233 117.828
2 (π
√
2;π
√
6) 3π
2 6.413 · 106 6.958 · 106
8
7 1 (0;π
√
2) π
2 50.118 722.124
2 (π
√
2;π
√
6) 3π
2 2.318 · 109 3.454 · 109
21
20 1 (0;π
√
2) π
2 2.302 · 106 91.789 · 106
2 (π
√
2;π
√
6) 3π
2 2.706 · 1026 11.159 · 1026
3 (π
√
6;π
√
12) 5π
2 5.676 · 1035 8.348 · 1035
41
40 1 (0;π
√
2) π
2 8.151 · 1013 644.211 · 1013
2 (π
√
2;π
√
6) 3π
2 1.165 · 1053 9.522 · 1053
3 (π
√
6;π
√
12) 5π
2 1.829 · 1071 5.329 · 1071
4 (π
√
12;π
√
20) 7π
2 1.761 · 1083 2.611 · 1083
40
41 1 (0;π
√
2) π
2 1.379 · 10−18 109.800 · 10−18
2 (π
√
2;π
√
6) 3π
2 1.003 · 10−56 8.254 · 10−56
3 (π
√
6;π
√
12) 5π
2 1.809 · 10−74 5.309 · 10−74
4 (π
√
12;π
√
20) 7π
2 3.701 · 10−86 5.528 · 10−86
20
21 1 (0;π
√
2) π
2 1.907 · 10−10 77.065 · 10−10
2 (π
√
2;π
√
6) 3π
2 1.645 · 10−29 7.043 · 10−29
3 (π
√
6;π
√
12) 5π
2 2.274 · 10−38 3.389 · 10−38
7
8 1 (0;π
√
2) π
2 6.552 · 10−5 97.983 · 10−5
2 (π
√
2;π
√
6) 3π
2 1.472 · 10−11 2.276 · 10−11
6
7 0 (0;π
√
2) π
2 1.866 · 10−4 24.247 · 10−4
1 (π
√
2;π
√
6) 3π
2 3.772 · 10−10 5.069 · 10−10
5
6 1 (0;π
√
2) π
2 0.001 0.006
2 (π
√
2;π
√
6) 3π
2 9.903 · 10−9 11.302 · 10−9
4
5 1 (0;π
√
2) π
2 0.002 0.015
3
4 1 (0;π
√
2) π
2 0.005 0.037
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
124 N. SVEIKATE, F. SADYRBAEV
p i Nonresonant intervals k ΓkMk Nk
2
3 1 (0;π
√
2) π
2 0.018 0.093
1
2 1 (0;π
√
2) π
2 0.075 0.239
Example. Consider the BVP
x′′ +
(π
2
)2
x = −|x|
6
5 signx, x′(0) = x(1) = 0. (6.4)
The function −|x|
6
5 signx is odd. And for x > 0 we can rewrite
x′′ +
(π
2
)2
x = −x
6
5 . (6.5)
Rewrite (6.5) equivalently
x′′ +
(π
2
)2
x+
(π
2
)2
x =
(π
2
)2
x− x
6
5 . (6.6)
The linear part in (6.6) is no more resonant with respect to (6.2).
We would like to make the function f(x) :=
π2
4
x − x
6
5 bounded and still continuous.
The function f(x) is an odd function with a maximum at x =
3125
7962624
π10. Define M :=
:= f
(
3125
7962624
π10
)
=
3125
191102976
π12. Solve the equation f(x) = −M for x > 0. The soluti-
on is N := 117.828.
Define the truncated function
F (x) =
− 3125
191102976
π12, x > 117, 828,
π2
4
x− x
6
5 , −117, 828 ≤ x ≤ 117, 828,
3125
191102976
π12, x < −117.828.
The function F (x) is continuous and bounded by the number M. Therefore the quasilinear
problem
x′′ +
π2
2
x = F (x), x′(0) = 0, x(1) = 0, (6.7)
has a solution x(t). Let us show that |x(t)| ≤ N ∀t ∈ [0, 1] and hence x(t) is also a solution of
problem (6.1), (6.2). A solution x(t) satisfies the integral equation
x(t) =
1∫
0
G(t, s)F (x(s)) ds, (6.8)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
QUASILINEARIZATION FOR RESONANT BOUNDARY-VALUE PROBLEMS . . . 125
(a) (b)
Fig. 6.1. 1-Type solution of the problem (6.4): (a) the trivial solution of problem (6.4), (b) the di-
fference between the neighboring solution and the trivial solution of problem (6.4).
(a) (b)
Fig. 6.2. 2-Type solution of problem (6.4): (a) the solution of problem (6.4) given for the initial data
ε′2(0) = 0, ε2(0) = 4000000, (b) the difference between the neighboring solution and the
solution ε2(t) of problem (6.4).
where G is Green’s function for the problem
x′′ +
π2
2
x = 0, x′(0) = 0, x(1) = 0.
It follows from (6.8) that
|x(t)| ≤ Γ ·M ∀t ∈ [0, 1],
where Γ =
√
2
π cos π√
2
, holds.
Then
x(t) ≤ N ∀t ∈ [0, 1]
satisfies, since
Γ ·M =
√
2
π cos π√
2
3125
191102976
π12 ≈ 11.233 < N ≈ 117.828.
This means that problem (6.1) allows for quasilinearization for k0 =
π
2
from the first nonreso-
nance interval (0;π
√
2), and the solution x(t) of the quasilinear BVP (6.7), (6.2) is also a soluti-
on of a original resonant BVP (6.1), (6.2). From the Main Theorem 3.1 the follows that it is
1-type solution Fig. 6.1. Problem (6.1), (6.2) has also a 2-type solution Fig. 6.2.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
126 N. SVEIKATE, F. SADYRBAEV
1. Agarwal R. P., O’Regan D. Ordinary and partial differential equations. — New York: Springer, 2009.
2. Conti R. Equazioni differenziali ordinarie quasilineari con condizioni lineari // Ann. mat. pura ed appl. —
1962. — 57. — P. 49 – 67.
3. Dobkevich M. On non-monotone approximation schemes for solutions of the second order differential
equations // Different. and Integral Equat. — 2013. — 26, № 9 – 10, — P. 1169 – 1178.
4. Elsgolc L. E. Differential equations and calculus of variations. — Moscow: Nauka, 1969 (in Russian).
5. Tricomi F. G. Differential equations. — London: Blackie, 1961.
6. Yermachenko I., Sadyrbaev F. Types of solutions and multiplicity results for two-point nonlinear boundary
value problems // Nonlinear Anal. — 2005. — 63. — P. e1725 – e1735.
7. Yermachenko I., Sadyrbaev F. Quasilinearization and multiple solutions of the Emden – Fowler type equation
// Math. Modelling and Anal. — 2005. — 10, № 1. — P. 41 – 50.
Received 24.09.13,
after revision — 03.01.14
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
|